# Hypocenter

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Contents

## Definition

### Symbol-free definition

The hypocenter of a group is defined as the limit of the lower central series of the group, that is, the intersection of all members of the lower central series.

## Properties

### Monotonicity

*This subgroup-defining function is monotone, viz the image of any subgroup under this function is contained in the image of the whole group*

The hypocenter of any subgroup is contained in the hypocenter of the whole group.

### Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once

The hypocenter of the hypocenter is the hypocenter. The image cum fixed-points of this are the hypocentral groups.

### Quotient-idempotence

This subgroup-defining function is quotient-idempotent: taking the quotient of any group by the subgroup, gives a group where the subgroup-defining function yields the trivial subgroup

View a complete list of such subgroup-defining functions

The hypocenter of the quotient of a group by its hypocenter is trivial.

## Properties satisfied

The hypocenter of any group is a fully characteristic subgroup.